Intrinsic topological superconductivity with exactly flat surface bands in the quasi-one-dimensional A$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs) superconductors

A spin-U(1)-symmetry protected momentum-dependent integer-$Z$-valued topological invariant is proposed to time-reversal-invariant (TRI) superconductivity (SC) whose nonzero value will lead to exactly flat surface band(s). The theory is applied to the weakly spin-orbit coupled quasi-1D A$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs) superconductors family with $p_z$-wave pairing in the $S_z=0$ channel. It's found that up to the leading atomic spin-orbit-coupling (SOC), the whole (001) surface Brillouin zone is covered with exactly-flat surface bands, with some regime hosting three flat bands and the remaining part hosting two. Such exactly-flat surface bands will lead to very sharp zero-bias conductance peak in the scanning tunneling microscopic spectrum. When a tiny subleading spin-flipping SOC is considered, the surface bands will only be slightly split. The application of this theory can be generalized to other unconventional superconductors with weak SOC, particularly to those with mirror-reflection symmetry.

Introduction.-Topologicalsuperconductivity (TSC) has aroused great research interests in the past decades, partly due to its potential application in the topological quantum computation [1,2].The key character of a topological superconductor lies in that its end (for 1D), edge (for 2D) or surface (for 3D) can accommodate gapless Majorana Fermions [1-4, 6-11, 13-16].In 1D, it's proposed that a semiconductor wire with Rashba spinorbit-coupling (SOC) in proximity to an s-wave superconductor can realize an effective p-wave TSC through a Zeeman coupling, which can accommodate Majorana end state [17].Such end state has been detected by the scanning tunneling microscope (STM) as a pronounced zero-bias conductance peak (ZBCP) [18].However, in 2D or 3D, the situation is subtle as the Majorana bands on the edge or surface generally have dispersion, which will broaden the bands and lead to weak [19][20][21] or no ZBCP [22,23] in the STM experiment.Therefore, the experimental identification of TSC in higher than 1D is still a challenge.
Here we investigate the evolution of the isolated end states of a 1D p-wave superconductor when a weak three dimensionality is exerted on the system to make it quasi-1D.Consequently, each isolated end state will be expanded into a branch of surface band.Generally, such surface band has dispersion, which hinders the STM detection.It's interesting to ask a question: is it possible to realize a quasi-1D TSC which hosts dispersionless surface band(s)?Under what symmetries are these flat surface bands topologically protected?Here we propose one possibility: the time-reversal (TR) combined with the spin-U(1) symmetries (SUS), under which we shall provide a momentum-dependent integer-Z-valued topological invariant, whose nonzero value will lead to exactly flat band(s) on the surface Brillouin zone.Differently from conventional nodal-line superconductors [9,[24][25][26][27][28], the flat surface band(s) here doesn't rely on the presence of the nodal line, and most times the surface band(s) can fill out the whole surface Brillouin zone, which will cause sharp ZBCP for the STM.Furthermore, we propose that the recently synthesized quasi-1D A 2 Cr 3 As 3 (A=Na, K, Rb, Cs) superconductors [29][30][31][32], which were predicted to have a p z -wave pairing symmetry [33,34], belong to this TSC class, up to the leading SOC.
In the theoretical aspect, symmetry analysis suggests that the leading SOC in the A 2 Cr 3 As 3 family is the atomic SOC which conserves the SUS [33], and that the possible triplet pairing states with line gap nodes only include the p z -and f -wave pairings [33,37].Considering the quasi-1D feature of this material, the p z -wave pairing is more possible.Both weak-coupling RPA and strongcoupling mean-field based calculations adopting realistic band structures have predicted TRI p z -wave triplet pair-ing with S z = 0 component (i.e.(↑↓ + ↓↑)) [33,34], conserving the SUS and is consistent with experiments.Therefore, the A 2 Cr 3 As 3 family are expected to belong to the symmetry class required here up to the leading SOC.
In this Letter, we provide topological invariant associated with flat surface band under combined TR+SUS symmetries for TSCs, and apply it to the quasi-1D A 2 Cr 3 As 3 superconductors family.As a result, the momentum (k x , k y )-dependent topological invariant Z(k x , k y ) for A 2 Cr 3 As 3 is nonzero all over the (k x , k y )plane, with some area covered by Z = 3 and others by Z = 2. Consequently, their surface spectrum exhibits exactly flat bands on the whole surface Brillouin zone, with different area hosting different numbers of flat bands.These flat bands will cause sharp ZBCP in the STM spectrum.When a tiny subleading spin-flipping SOC turns on, the surface flat bands will only be slightly dispersed, which will cause no obvious change in the STM spectrum.Our proposal provides smoking-gun evidence for experimental identification of such quasi-1D p z -wave SC as that in the A 2 Cr 3 As 3 family, which possess only weak SOC or dominant SOC with SUS.
SUS protected Topological invariant.-Let'sconsider a multi-band TRI superconductor with SUS, whose Bogolubov-de Gennes (BdG) Hamiltonian is k .This BdG Hamiltonian can be written in the particle-hole symmetric (PHS) 4-component Nambu representation as Here Note that the SUS requires the hopping blocks of the BdG matrix Eq.( 2) to be diagonal and the pairing blocks to be block offdiagonal.The combined TRS and PHS lead to chiral symmetry, which enables us to do the unitary transformation H → H = U † HU to obtain an off-diagonal Hermitian matrix H.Here the unitary matrix U and the upper-right off-diagonal block H 12 of H read (3) Note that H 12 is block-diagonal, caused by the SUS.
Following the standard procedure introduced in Ref [1,2,25], the so-called Q-matrix is obtained Here P (k) is the projection operator, and the N α × N α matrix q 1 (q 2 ) is related to the upper-left (lower-right) block of H 12 in Eq.( 3), and hence the 1-4 (2-3)-block of H in Eq. (2).Each block of the Hamiltonian only has the chiral symmetry and belongs to the AIII class, which is characterized by a Z topological invariant in 1D.The detailed derivation of the formulae of q 1/2 is provided in the Supplementary Material (SM) [49].Note that in contrast with the cases in ordinary TRI superconductor [1,2,25], the extra SUS here makes the off-diagonal block q ≡ diag(q 1 , q 2 ) of the Q matrix block-diagonalized into q 1 and q 2 sub-blocks.This enables us to define the following two 1D winding number Z 1/2 for the two sub-blocks q 1/2 , instead of the one Z defined for the whole q for ordinary TRI superconductors [1,2,25].
Note that due to the double counting brought about by the gauge redundancy in this representation, the physical topological invariant here should be Z 1 (k x , k y ), which leads to |Z 1 (k x , k y )| flat surface bands.[49] It's interesting to investigate the case of intrabandpairing limit where ∆ αβ k = ∆ α k δ αβ .In this case, it's obtained [49] that Then from Eq.( 5), the winding number is obtained as Eq.(6) suggests that in this limit, Z 1 (k x , k y ) is a summation of the contributions from each band α, with each contribution equal to the winding number of the complex phase angle of ε kα↑ + i∆ α k along a closed path perpendicular to the k z = 0 plane.
Applied to A 2 Cr 3 As 3 .-Thepoint group of the quasi-1D A 2 Cr 3 As 3 family is D 3h , which includes a C 3 -rotation about the z-axis and a mirror reflection about the xyplane.The low-energy band structure of A 2 Cr 3 As 3 can be well captured by the following three-band tightbinding (TB) model in the absence of SOC, Here µ/ν = 1, 2, 3 denotes the 3d z 2 , 3d xy and 3d x 2 −y 2 orbitals respectively.Below, we take the K 2 Cr 3 As 3 as an example, whose h µν (k) is provided in Ref [33].From symmetry analysis [49], the leading SOC in this family takes the following on-site formula [33], which possesses the SUS required.We adopt λ so ≈ 10meV [33] below.Note that the mirror-reflection symmetry forbids spin-flipping on-site SOC, as each such term as c † iµσ c iνσ g µν would be changed to c † iµσ c iνσ σσg µν = −c † iµσ c iνσ g µν under the mirror-reflection operation [49].
The FSs of the spin-up electrons shown in Fig. 1(a) consists of two 1D FSs named as α and β and one 3D FS named as γ.While each 1D FS contains two FS sheets nearly parallel to the (k x , k y )-plane, the 3D γ-FS intersects with the (k x , k y )-plane with their intersection line shown in Fig. 1(b).Note that the shape of the γ-FS is counter-intuitive: it contains one connected large concave pocket centering around the Γ-point, instead of three isolate small convex pockets centering around the M -points.The FSs of the spin-down electrons are related to those of the spin-up electrons through the relation ε kα↓ = ε −kα↑ brought about by the TRS, and the lack of inversion symmetry leads to ε −kα↑ = ε kα↑ and hence ε kα↓ = ε kα↑ , which means that the band structures of the two spin-species don't coincide.
Both weak-coupling RPA-based calculations and strong-coupling mean-field results suggest that the leading pairing symmetry of the system is TRI p z -wave pairing with a dominating triplet component in the S z = 0 channel with line nodes [33,34] consistent with experiment [29][30][31]44], conserving SUS.Therefore, the A 2 Cr 3 As 3 family are expected to belong to the symmetry class required here.Since the T c ( ≤ 8 K ) of A 2 Cr 3 As 3 is much lower than the low-energy band width (≈ 100meV), its pairing state can be well approximated as intra-band pairing, wherein Eq.( 6) applies.
The gap function of the p z -wave pairing obtained by the RPA approach is C 3 -rotation invariant about the z-axis and doesn't obviously depend on k x/y .The k zdependence of the relative gap function averaged on the FSs is shown in Fig. 1(c), where the sign of ∆ α k (α = 1, 2, 3) follows that of k z .Let's take the γ-band as an example to evaluate how to use Eq.( 6) to calculate I γ .Figure 1(d) and (e) illustrate in a schematic manner how the complex phase angle of ε kγ↑ +i∆ γ k evaluates from the A i to E i points along the two vertical lines L i(i=1,2) in the Brillouin zone shown in Fig. 1(a).Clearly, because the L 1 path passes the γ-FS twice which leads to twice sign changes of ε kγ↑ and that the p z -symmetry leads to sign change of ∆ γ k on thetwo γ-FS sheets, a nontrivial winding number I γ (L 1 ) = −1 of the phase angle is obtained.On the contrary, the L 2 path doesn't pass the γ-FS, which leads to no sign change of ε kγ↑ and hence The pairing state in K 2 Cr 3 As 3 can be approximately generated by the following MF Hamiltonian including only NN intra-orbital pairing potential [33], This pairing state has a weak inter-band pairing component and its intra-band pairing component is well consistent with that obtained by the RPA approach.The topological invariants Eq.( 5) for this pairing state yield exactly the same results as those of the RPA.It's remarkable that the topological properties studied here are protected by the SUS.Supposing a vanishingly weak spin-flipping SOC turns on, the system now belongs to conventional TRI superconductors, whose topological invariant Z is then defined for the whole off-diagonal block q of Eq.( 4) [1,2,25] In the case with weak on-site SOC with SUS for the p z -wave SC, we have (6) we find that except in a narrow regime to be studied below, in most regime of the (k x , k y )-plane we have Z 1 (k x , k y ) = −Z 1 (−k x , −k y ) for integer Z 1 , and hence Z(k x , k y ) = 0.Here the protection of the SUS permits that we only count Z 1 , which is nonzero all over the (k x , k y )-plane.Surface spectrum and STM.-The nontrivial topological invariant of A 2 Cr 3 As 3 leads to flat surface bands.Here we study the spectrum of Eq.( 9) with open boundary condition along the z-axis and periodic ones along the x-and y-axes [49].The obtained energy spectrum as function of k x and k y is shown in Fig. A28(a) along the high symmetric line, in comparison with the bulk band in the superconducting state shown in Fig. A28(b) with fixed k z = 0. To enhance visibility, the adopted pairing gap amplitudes are enhanced from realistic ∆ The comparison between Fig. A28(a) and (b) suggests that, in addition to the bulk continuum, some regime in the (k x , k y )-plane is covered by extra 6 flat bands while the remaining regime is covered by extra 4, with the boundary of the two regimes to be just the SC nodal lines shown in Fig. 1(b).
The flat bands shown in Fig. A28(a) are formed by bound states localized at the two (001) surfaces, which is justified by the distribution of the wave functions of the Bogolubov quasi-particles shown in Fig. A32(a), which illustrates a bound state with a localized length ξ ≈ 3c with the lattice constant c = 4.23 Å.Our numerical results suggest ξ ≈ 45c ≈ 20 nm for realistic gap amplitudes.As the two (001) surfaces symmetricly share the surface states, the corresponding areas in Fig. A28(a) are covered by 3 (2) flat bands on each surface Brillouin zone, consistent with the topological invariant calculations above.
The topological flat surface bands obtained above can be detected as the ZBCP in the site-dependent differential conductance spectrum dI/dV of the STM.Details for the calculation of dI/dV are provided in the SM [49].whose explicit formula is given in SM [49].This SOC term breaks the SUS, and the topological invariant Eq.( 5) doesn't apply.However, as this NN-SOC for the 3dorbitals is so weak (with strength λ 1/2 = 2 meV adopted) that the above obtained flat bands are only slightly split, as shown in Fig. 4(a) and its zoom-in in Fig. 4(b).As a result, the sharp ZBCP is still present in the STM spectrum shown in Fig. 4(d).
It's remarkable that even the spin-flipping SOC breaks the SUS here, there is still a narrow regime in the Brillouin zone covered by a pair of exactly-flat bands, as highlighted by the red oval in Fig. 4(b).This pair of exactly-flat bands are protected by the topological invariant Z for conventional TRI SC without SUS [1,2,25].As introduced above, for sufficiently weak λ Ra , Z(k x , k y ) = Z 1 (k x , k y ) + Z 1 (−k x , −k y ).As the A 2 Cr 3 As 3 family is noncentrosymmetric, the weak difference between ε kα↑ and ε −kα↑ caused by Eq.( 8) leads to weak noncentrosymmetry in |Z 1 (k x , k y )| and hence nonzero Z(k x , k y ) = ±1 in the narrow shaded regime in Fig. 4(c), which causes a pair of exactly-flat surface bands there.
Discussion and Conclusion.-One may worry about the difficulty in the detection of the surface states on the narrow (001) surface which can hardly be a cleavage plane.What's more, there exists the problem that the breaking of mirror-reflection symmetry on the surface might lead to on-site spin-flipping SOC on the few surface layers, influencing the surface states there.These two problems, however, can be easily solved by noticing that the surface state actually has a considerably long localized length of ξ ≈ 45c ≈ 20 nm introduced above.As shown in Fig. 4(e), when the tip of the STM is put on the side surface near the end of the sample within the localized length, there would be pronounced ZBCP in the STM spectrum, and when the tip is far from the end the ZBCP would be absent.Note that there is no extra flat bands on the side surface of the sample.
It's interesting that the FS topology of A 2 Cr 3 As 3 can be drastically changed upon slightly doping, accompanied by several Lifshitz transitions [49].As a result, the distribution and the number of topological flat surface bands will be easily engineered through doping, which can be detected by experiments.The SUS protected topological invariant proposed here will not only apply to the A 2 Cr 3 As 3 family, but it will also apply to a wide class of unconventional superconductors with weak SOC, particularly those with mirror-reflection symmetry.This symmetry will forbid spinflipping term in the leading (on-site) SOC [49], causing the SUS required here.
In conclusion, we have proposed an SUS protected momentum-dependent integer-Z-valued topological invariant for TRI superconductors, whose nonzero value will lead to exactly-flat surface bands.When this theory is applied to the A 2 Cr 3 As 3 family up to the leading SOC, it's obtained that the whole surface Brillouin zone on the (001) surface is covered with exactly-flat surface bands, with different regimes hosting different numbers of flat bands.These flat bands can be detected by the STM as sharp ZBCP.When the weak sub-leading spinflipping SOC turns on, besides the presence of the almost flat bands with only slight split, there remains a narrow regime in the surface Brilloiun zone which is covered by exactly-flat bands.The divergent DOS brought about by the flat bands on the surface might probably cause exotic electron instabilities other than the bulk SC under electron-electron interactions.We leave such topic for future study.The triplet pairing with S z = 0component predicted here should be identified by the NMR experiment through the Knight-shift measurement.Note that the other pairing-symmetry candidate, i.e. the f -wave [37], will exhibit linear-energy-dependent DOS on the (001) surface, in sharp contrast to the p z -wave state.Our discovery not only reveals a new type of TRI TSC, but it also provides smoking gun evidence for the experimental identification of the p z -wave pairing symmetry of the quasi-1D A 2 Cr 3 As 3 family.
In this section, we derive the spin-U(1)-symmetry (SUS) protected momentum-dependent integer-Z-valued topological invariant for time-reversal-invariant (TRI) superconductors, whose nonzero value will lead to exactly flat band(s) on the surface Brillouin zone.
We start from the following N α -band model: Here α/β = 1, • • • , N α denote the band indices.Note that here we have allowed SOC with SUS and inter-band pairing.From time-reversal symmetry (TRS), we obtain ε α k↑ = ε α −k↓ and ∆(k) = ∆ † (k).We can rewrite H into the formula of where and Note that with Let's study the eigenvalues and eigenvectors of H k and H k .It's proved here that for the fully-gapped case of H k , the 1-2 (3-4) diagonal block contributes N α negative eigenvalues and N α positive ones respectively, with the corresponding eigenvectors in the form of µ T , ν T , 0, 0 T and µ T , −ν T , 0, 0 T ( 0, 0, µ T , ν T T and 0, 0, µ T , −ν T T ).
Here µ and ν are N α -component column vector.Actually, to solve the eigenvalue problem of the 1-2 diagonal block of H k in Eq.(A7), one needs to solve the equation with λ to be the eigenvalue.Since we have det(F † F − λ 2 I) = 0.Because F † F is a Hermitian operator with positive-definite eigenvalues, the obtained values of λ are always positive-negative symmetrically distributed.Therefore, the 1-2 diagonal block of the H k in Eq.(A7) contributes N α negative eigenvalues and N α positive ones respectively, with the corresponding eigenvectors in the form of µ T , ν T , 0, 0 T and µ T , −ν T , 0, 0 T .Similarly, the 3-4 diagonal block of the H k in Eq.(A7) contributes the same number of negative and positive eigenvalues, with the corresponding eigenvectors in the form of 0, 0, µ T , ν T T and 0, 0, µ T , −ν T T .Then from Eq.(A6), the distribution of the eigenvalues of Hk is similar, but with corresponding eigenvectors in the form of µ T , 0, ν T , 0 T and µ T , 0, −ν T , 0 T ( 0, µ T , 0, ν T T and 0, µ T , 0, −ν T T ).
Following the standard procedure for the topological invariant of TRI SC [1,2], we evaluate the projection operator P as are the 2N α eigenvectors of Eq.(A5) with negative eigenvalues: The formulae of q 1 , q 2 are Here we have used the relation Note that Q is 2 × 2 block-off-diagonal, and furthermore the off-diagonal block is block-diagonal.Due to this property, the winding number can be defined as Z 1/2 , with For nontrivial Z 1/2 (k x , k y ) = 0, there will be |Z 1 |+|Z 2 | zero-modes at the momentum (k x , k y ) on the surface Brillouin zone in the above 4-component Nambu representation (A2).However, the gauge redundancy in this representation brings up double counting on the number of zero-modes for each momentum.In fact, due to the SUS here, one can also write the BdG Hamiltonian in the gauge-redundancy-free 2-component Nambu-representation as and h is equal to the 1-4 block of Eq.(A3).In this representation, the number of zero-modes at momentum (k x , k y ) is |Z 1 (k x , k y )|, and the other |Z 2 (k x , k y )| zero-modes obtained in the 4-component representation are just those folded from the momentum (−k x , −k y ) artificially in that representation, which is a double counting.Therefore, the physical topological invariant here is Z 1 (k x , k y ), which leads to |Z 1 (k x , k y )| flat surface bands.
In the following, we consider the special case of intra-band pairing limit, which is the case of most of the existing superconductors.Let's set ∆ αβ (k) = ∆ α (k)δ αβ and perform the calculations provided by Eq.(A5), Eq.(A11) and Eq.(A12).As a result, we obtain A. Formula of spin-U(1)-symmetric SOC In this subsection, we analyze possible SOC terms with SUS, including the on-site formulae in 1 and the NN ones in 2.

Formula of on-site SOC conserving SUS
The on-site SOC with SUS can generally be written as with the g-matrix to be Hermitian.Note that the TRS has been considered here.Let's evaluate the requirement on g µν by the C 1 3 -rotation symmetry for the spin-up electrons.
µµ D (1) 1) † gD (1) ) µν .To keep rotation symmetry, we have Solving this equation, we get the symmetry-allowed formula for the 3 × 3 Hermitian matrix g as Note that the weak diagonal term can be incorporated into the on-site chemical potential term, which will be ignored here.As a result, we get the Hamiltonian term describing this SOC, H It can be checked that this Hamiltonian satisfies all the symmetries listed above.

Formula of NN-SOC with SUS
The combined TRS represented by Eq.(A18), the mirror symmetry represented by Eq.(A19) and the Hermitian character of the Hamiltonian require that the NN-SOC conserving the SUS takes the formula of with the g-matrix Hermitian.Then the C 1 3 -rotation symmetry represented by Eq.(A21) requires the same formula Eq.(A25), which is solved as Eq.(A26).Again, the weak diagonal part has extra spin-SU(2)-symmetry and can be incorporated into the band structure part, which will be ignored here.Therefore, we obtain B. Formula of spin-flipping SOC In this subsection, we analyze possible spin-flipping SOC terms breaking SUS, including the on-site formulae in 1 and the NN ones in 2.
1. Formula of spin-flipping on-site SOC It's proved here that the mirror-reflection symmetry M will forbid spin-flipping on-site SOC.Actually, assume we have a spin-flipping SOC term in the form of C † iµσ C iνσ g µν with σ =↑ or ↓, then from the mirror symmetry M about the plane passing through i, we have: Since M should be respected, we have g µν = 0, which suggests that the mirror-reflection symmetry M will forbid spin-flipping on-site SOC. the iterative Green's function approach for K2Cr3As3 with SUS, respectively.The adopted ∆1=20 meV, ∆2 = ∆3=40 meV are enhanced by an order of magnitude over realistic ones to enhance the visibility.The on-site SUS SOC λso = 10 meV.

Formula of spin-flipping NN-SOC
Here we evaluate the possible NN-spin-flipping SOC term.From the combination of the TRS represented by Eq.(A18), the mirror symmetry represented by Eq.(A19) and the Hermitian character of the Hamiltonian, such SOC term takes the following general formula, with g µν = g νµ .This Hamiltonian is already Hermitian.Then the C 1 3 -rotation symmetry represented by Eq.(A21) leads to the equation, D † gD (1) = ge This equation can be solved as where λ 1 and λ 2 are two independent coupling constants.Therefore, the symmetry-allowed NN-spin-flipping SOC term in the A 2 Cr 3 As 3 family takes the form of Eq.(A29), with the symmetric g-matrix provided by Eq.(A31).

III. SURFACE SPECTRUM
In the following, we adopt two different approaches to calculate the surface spectrum of the system in the presence of SUS or Rashba SOC.In the first approach, we directly diagonalize the Bogoliubov-de Genes Hamiltonian of the superconducting system using a slab geometry with open boundary condition along the z-axis (the width is 200c) and periodic ones along the x-or y-axis.In the second approach, we utilize an iterative method [3] to obtain the surface Green's functions of semi-infinite systems, from which we calculate the dispersions of the surface states.The results obtained from both approaches agree well with each other, which exhibit the exactly flat surface spectrums shown in Fig. A1 and Fig. A2.

IV. STM
The site-dependent differential conductance dI/dV spectrum of the STM can be evaluated as ρ (i z , ω) = ρ (i x , i y , i z , ω) (A32) = −Im where N x , N y are the size of lattice along x and y direction, E m (k x , k y ) is the m-th eigenvalue of the BdG Hamiltonian matrix for the fixed momentum (k x , k y ) and ψ izµ,m (k x , k y ) is the corresponding eigenvector.Note that the spectrum only depends on i z and not on i x or i y .The coordinates i z = 1(N z ) and i z = N z /2 correspond to the end and the middle of the sample, respectively.

V. FERMI SURFACE EVOLUTION AND LIFSHITZ TRANSITION UPON DOPING
The FS topology of K 2 Cr 3 As 3 can be drastically changed upon slightly doping, accompanied by several Lifshitz transitions.As a result, the distribution of the number of topological flat surface bands will be easily engineered through doping, which can be detected by experiments.The FSs in both Figures ( Fig.

FIG. 1 .
FIG. 1. (Color online) FSs for the spin-up electrons and pairing gap function of K2Cr3As3.(a) FSs of the TB model (7) + (8) for K2Cr3As3.The paths Li,i=1,2 are perpendicular to the (kx, ky)-plane.(b) The intersection lines between the γ-FS and the (kx, ky)-plane, which are also the nodal lines of the pz-SC.(c) The kz-dependence of the relative gap function of K2Cr3As3 averaged on the FSs.Schematic diagrams of how the phase angle of ε kγ↑ + i∆ γ k evolve with kz along the paths L1 for (d) and L2 for (e).
Therefore, the three elliptical areas centered around the M points (the remaining part) in Fig. 1(b) are covered by Z = −2 (−3), which will lead to 2 (3) flat bands over this regime on the surface Brillouin zone on each (001) surface.

FIG. 2 .
FIG. 2. (Color online) (a) The spectrum of Eq.(9) as function of k x/y with open boundary condition along the z-axis and periodic ones along the x-and y-axes.(b) The bulk bands in the pz-wave pairing state with fixed kz = 0 for Eq.(9).In (a), the segment marked red (blue) is covered by 6 (4) flat bands.The number of the slab layers is 200.The adopted ∆1=20 meV, ∆2 = ∆3=40 meV are enhanced by an order of magnitude over realistic ones to enhance the visibility.

FIG. 3 .
FIG. 3. (Color online) (a) Distribution of the squared modulus of the wave function of the particle part of the Bogolubov quasi-particle (the hole-part is similar) along the z direction for a typical state in the flat bands.(b) The differential conductance dI/dV ∼ V spectra of the STM for the end (red) and the middle (black) of K2Cr3As3 sample .

FIG. 4 .
FIG. 4. (Color online) The surface spectrum (a) and its zoom-in (b) in the presence of the NN-spin-flipping SOC with strength λ1 = λ2 = 2 meV for K2Cr3As3.(c) The noncentrosymmetric distribution of Z1(kx, ky) in the surface Brillouin zone, with the narrow shaded regime covered by |Z| = 1.(d) The STM spectra for the end (red) and the middle (black) of the sample respectively.(e) The schematic configuration for experimental identification of the pz-wave SC in the quasi-1D A2Cr3As3 family.
FIG. A1. (Color online) The surface spectra obtained from (a) diagonalizing the BdG Hamiltonian directly using a slab geometry with open boundary condition along the z-axis (the width is 200c) and periodic ones along the x-or y-axis and (b)the iterative Green's function approach for K2Cr3As3 with SUS, respectively.The adopted ∆1=20 meV, ∆2 = ∆3=40 meV are enhanced by an order of magnitude over realistic ones to enhance the visibility.The on-site SUS SOC λso = 10 meV.

2 ω 2 ω 2 ω
kxkyiz,µσ G e −i(kx ix+ky iy ) √NxNy − E + i0 + = − 2Im N x N y E kx,ky,µ,m E γ † kx,ky,m G ψ * izµ,m (k x , k y ) − E + i0 + = − 2Im N x N y kx,ky,m,µ |ψ izµ,m (k x , k y )| − E m (k x , k y ) + i0 + ,(A33) A3 for hole doping and Fig. A4 for electron doping) are the FSs of the spin-up electrons, and the FSs for the spin down channel can be obtained by time-reversal operation.Consistent with the topological invariant calculations in the main text, the corresponding areas in the surface spectrum of Fig. A3 and Fig. A4 are covered by 2 or 3 flat bands on the surface Brillouin zone.